I'm definitely messing up somewhere :P(adsbygoogle = window.adsbygoogle || []).push({});

The Larmor Equation is

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

I derived an equation stated as

[tex]\nabla \times \vec{F}_{ij} \cdot \hat{n} = \nabla \times \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

How is put in a simpler way from the original derivation, is imagine we have what we began with

[tex]\vec{F}_{ij} = \frac{\partial V(r_{ij})}{\partial r_{ij}} \hat{n}[/tex]

Take the dot product of the unit vector on both sides gives

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

Now taking the curl of F is

[tex]\nabla \times \vec{F}_{ij} \cdot \hat{n} = \nabla \times \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

In light of this equation however

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

I could rewrite the Larmor energy as

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e}(\vec{F}_{ij} \cdot \hat{n}) L \cdot S[/tex]

But I am sure many agree that is not very interesting.

(and this is really starting to make my brain cells burst), taking the curl of a force gives a F/length, however, the unit vector in this equation

[tex]\vec{F}_{ij} \cdot \hat{n} = \frac{\partial V(r_{ij})}{\partial r_{ij}}[/tex]

cancels these out and what I have again is the force again... yes? ... very circular... since this would be true, then we know what the force is anyway given earlier:

[tex]\vec{F}_{ij} = \frac{\partial V(r_{ij})}{\partial r_{ij}} \hat{n} [/tex]

Now, just a moment ago I found the Larmor energy written as

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{1}{r}\frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

Notice the 1/r term which is not in my original case. If that where true, and we plug in my force example again

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{1}{r}\frac{\partial V(r_{ij})}{\partial r_{ij}}\hat{n} ( L \cdot S)[/tex]

(If I am doing this right) the unit vector would cancel out with the radius term and what would be left with is

[tex]\Delta H = \frac{2\mu}{\hbar Mc^2 e} \frac{\partial V(r_{ij})}{\partial r_{ij}} ( L \cdot S)[/tex]

Now if that wasn't confusing I don't know what is.

I've either had it right from the beginning, or I have made a tiny mistake which is making a huge impact on my understanding of my own equation. Any help would be gladly appreciated.

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# Can you help me solve my mathematical problem?

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